Tapping an air cored RF inductor

Introduction

A question recently asked in an online forum was:

Since I built my first 80meter/40meter 6aq5 + 6DQ6 transmitter
with
pi
output in 1972, when I want to vary the inductance of a coil in a
tunner, or loading coil in an antenna, I just short circuit some turns.

I see that this is the usual practice everywhere.

My question is why do we not just leave the turns open circuited
instead of short circuiting them.

It appears to me that in the short circuited turns, a very big
current
must be circulating, adding heat losses and lowering the Q of the
circuit.

It is an interesting question, and one that is easily explained by
basic AC circuit theory.

Leaving turns open circuit

The first question was why not leave the unused turns open circuit.

The reason is that the voltage induced in the unused turns can be
very high, extreme in the case when tapped close to the other end, and
a challenge for switch insulation, coil insulation etc.

The induced voltage can be calculated easily using basic AC circuit
theory.

A model for loss in shorted turns

A simple model can be constructing by making some assumptions:

inductance of an air cored solenoid is reasonably estimated by
Wheeler's formula;

distributed capacitance is insignificant;

the current in each section of the inductor is uniform;

loss is proportional to the length of wire times current squared.

Wheeler's formula

Wheeler's formula estimates the inductance of an air cored single
layer solenoid as L=µ0*π*N^2*r^2/(l+0.9*r) where L is in H, N is number
of turns, r is the radius of the solenoid in metres, l is the height of
the solenoid in metres.

Wheeler later published his continuous inductance formula,
L=
µ0*N^2*r*(ln(1+pi*r/l)+1/(2.3004+1.6*l/r+0.4409*(l/r)^2)), which is
more accurate for the short inductors in this analysis.

Technique

The technique is to choose a practical example coil, and for a range
of tapping positions, to calculate inductance of each section of coil L1
and L2 as an independent inductors using Wheelers
formula, and knowing (from Wheelers formula) the inductance of the
coupled combination, calculate the mutual inductance M between the
sections.

From L1, L2 and M, we can calculate form a T
equivalent circuit for the coupled coil sections, and calculate the
current in the shorted turns relative to the current in the other part
of the inductor.

Assuming that loss resistance is proportional to the length of wire,
the relative loss in the shorted turns can be calculated.

Results

Fig 1:

Fig 1 shows the loss in the shorted turns as a percentage of total
loss for an air cored inductor of 20 turns, 50mm diameter, and various
winding pitches (metres per turn).

It can be seen that for closer spaced windings (smaller pitch), loss
is higher. The inductor sections are more tightly coupled, higher flux
linkage, more current flows in the shorted turns, and loss is higher.

Fig 2:

Fig 2 shows the same data with an expanded scale. It can be seen
that loosely coupled turns give rise to lower loss, and avoiding
shorting few turns helps to keep losses lower.

The flux coupling factor k for the blue line is about 0.47. Higher
coupling factors, as might be obtained using a magnetic core, will
result in higher relative loss in the shorted turns.

Note that the model shorts only the tapping point to the coil end,
and intermediate tap points are left open.

So, in answer to the question, yes a very large circulating current
can flow in the shorted turns. The circulating current is highest for
tightly coupled turns, and when very few turns are shorted. In this
situation, resistance of the shorted turns is relatively low, but
current square may be relatively high. Fortunately, many applications do
not call for shorting very few turns.

Note that shorting just one end turn has a dramatic effect on
inductance when the coupling factor is high, so not only is it
potentially lossy, the reduction in inductance may be more than
desired. Lower coupling factor tames this behaviour, but at the expense
of a physically larger inductor.

Conclusions

Tapping a coil and leaving unused turns open can create high
voltages in the unused turns, worst when the coil is tightly coupled and
the tapping point is near to the other end of the coil.

Shorting unused turns is practical on coils where flux coupling is
not high, and losses are lowest when tapping points that give rise to
very few shorted turns are avoided.